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FOUNDBD  BY  JOHN  D.  ROCKEFELLER 


DEVELOPMENT  OF  THE  PROFILE  OF 
EQUILIBRIUM  OF  THE  SUBAQUEOUS 
SHORE  TERRACE 


A    DISSERTATION    SUBMITTED    TO    THE    FACULTIES    OF    THE    GRADUA'J 
SCHOOLS    OF   ARTS,   LITERATURE,   AND    SCIENCE,    IN    CANDIDACY 
FOR    THE    DEGREE    OF    DOCTOR    O?"  PHILOSOPHY 

(department  of  geology) 


BY 

N.  M.  FENNEMAN 


PRINTED    BY 

Zbc  'dntpetdits  ot  Cbicaao  t>tes9 

CHICAGO 


Ft 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/developmentofproOOfennrich 


DEVELOPMENT  OF  THE  PROFILE  OF  EQUILIBRIUM 
OF  THE  SUBAQUEOUS  SHORE  TERRACE. 


The  profile  of  a  shore  as  seen  at  any  one  time  is  a  compro- 
mise between  two  forms.  One  of  these  is  the  form  which  it 
possessed  when  the  water  assumed  its  present  level ;  from  this 
form  it  is  continually  departing.  The  other  is  the  form  which 
the  water  is  striving  to  give  to  it ;  toward  this  form  it  is  continu- 
ally tending.  There  is  a  profile  of  equilibrium  which  the  water 
would  ultimately  impart,  if  allowed  to  carry  its  work  to  comple- 
tion. The  continual  change  of  shore  line  and  the  supply  of  new 
drift  are  everchanging  conditions  with  which  no  fixed  form  can 
be  in  equilibrium.  There  are,  however,  certain  adjustments  of 
current,  slope  and  load  which,  when  once  attained,  are  maintained 
with  some  constancy.  The  form  involved  in  these  adjustments 
is  commonly  known  as  the  profile  of  equilibrium.  When  this  pro- 
file has  once  been  assumed  the  entire  form  may  slowly  shift  its 
position  toward  or  from  the  land,  but  its  slope  will  change  little 
or  not  at  all.  It  may  be  compared  to  a  stream  channel  which 
has  reached  grade  but  not  base  level. 

The  force  which  the  water  exerts  is  derived  ultimately  from 
the  wind.  The  immediate  agencies  in  the  work  are  waves  and 
currents.  It  will  be  convenient  to  consider  these  first  as  acting 
independently  of  the  wind  which  caused  them,  and  second,  as 
acting  under  its  continuous   influence.     It   is  also   desirable  to 


13331H 


2  N.  M.  FENNEMAN 

consider  waves  first  in  their  free  forms,  while  meeting  no  resist- 
ance and  hence  doing  no  external  work.  This  condition  is  found 
in  deep  water.  The  various  ways  in  which  the  bottom  or  shore 
may  offer  resistance  and  be  subject  to  work  may  then  be  dis- 
cussed. 

WAVES    IN    WATER    OF    INFINITE    DEPTH  . 

When  wave  agitation  does  not  reach  the  bottom  of  a  body  of 
water  it  is  customary  to  speak  of  the  depth  as  infinite,  because 
the  wave  is  not  influenced  by  the  existence  of  a  bottom. 

PURE   OSCILLATION. 

Orbits. —  In  simple  oscillatory  waves  each  water  particle 
moves  in  a  circular  and  closed  orbit.  The  water  body  itself, 
therefore,  has  no  onward  motion.  These  orbits  diminish  rapidly 
with  depth,  but  so  long  as  agitation  does  not  reach  the  bottom, 
the  orbits  are  circles  at  all  depths.*  The  particles  on  the  crest 
are  moving  in  the  direction  in  which  the  wave  is  traveling  and 
particles  in  the  trough  are  moving  with  the  same  velocity  in  the 
opposite  direction. 

Difierential  movement. — -On  a  line  in  the  direction  of  the  wave 
movement  (hence  crossing  the  waves  at  right  angles)  each  par- 
ticle is  subject  to  a  gliding  between  its  neighbors.  The  amount 
of  this  gliding  is  of  molecular  dimensions,  hence  not  infinitely 
small.  It  will  be  spoken  of  here  as  the  differential  movement  of 
particles.  For  diagrammatic  purposes  it  is  convenient  to  consider 
this  differential  as  a  considerable  arc  of  the  orbit,  hence  particles 
are  chosen  which  are  removed  from  one  another  by  a  consider- 
able fraction  of  the  length  of  the  diameter. 

General  form  of  wave. —  In  a  series  of  particles  moving  in 
equal  orbits  each  particle  is  more  advanced  in  its  orbit  (has  a 
more  advanced  phase)  than  the  one  in  front  of  it.     If  a  series 

*This  principle  was  clearly  elucidated  by  Gerstner  in  1804.  This  and  other  fun- 
damental principles  of  oscillatory  wave  motion  are  clearly  set  forth  in  the  report  of  the 
brothers  Weber  on  their  experiments  conducted  in  the  early  part  of  the  last  century. 
See  "  Wellenlehre,  Ernst  Heinrich  und  Wilhelm  Weber,  Leipzic,  1825.  This 
report  also  summarizes  Gerstner's  work  and  all  other  previous  studies  on  waves  from 
the  time  of  Newton  to  1820. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE         3 

of  orbits  be  drawn  and  the  positions  of  the  several  particles  be 
connected  by  a  curved  line,  that  line  will  show  the  wave  form 
(Figs.  I  to  6).  The  curve  is  a  trochoid/  It  may  be  produced 
by  rolling  a  circle  on  the  under  side  of  a  straight  horizontal  line. 

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^  First  recognized  by  Gerstner,  Iheorie  der  Wellen,  Prague,  1804.  See  p.  343 
of  reprint  in  Weber's  Wellenlehre.  Discussed  mathematically  by  W.  J.  M.  Rankine, 
"  On  the  Exact  Form  and  Motion  of  Waves  at  and  near  the  Surface  of  Deep  Water," 
Philosophical  Transactions,  1863,  page  127. 


4  N.  M.  FENNEMAN 

The  path  generated  by  any  point  within  the  circle  is  a  trochoid. 
This  line  will  be  sharply  curved  or  broadly  curved,  approaching 
straightness,  according  as  the  point  which  generates  it  is  chosen 
near  the  circumference  or  near  the  center.  The  distance  from 
the  center  to  the  generating  point  is  called  the  tracing  arm. 
When  the  point  is  at  the  circumference — that  is,  when  the  trac- 
ing arm  equals  the  radius  of  the  rolling  circle  —  the  curve  is 
cusped  at  the  top  and  is  the  common  cycloid  (see  Fig.  8).  This 
is  the  steepest  and  shortest  form  which  a  true  wave  can  have. 
If  the  tracing  arm  be  longer  that  the  radius  of  the  moving  circle 
the  curve  is  looped  instead  of  cusped  (^Fig.  4).  The  failure  of 
the  water  surface  to  assume  these  looped  forms  results  in 
breakes. 

Steepness  dependent  on  differential  movement. —  If  the  same  series 
of  particles  in  their  orbits  be  represented  in  several  diagrams, 
assuming  for  each  diagram  a  different  amount  of  differential 
movement,  the  wave  will  be  found  to  be  long  when  the  differen- 
tial movement  is  small,  and  short  when  the  differential  movement 
is  large  (compare  Figs,  i  and  2).  If  the  size  of  the  orbits  be 
increased  while  the  distance  between  the  particles  remains  the 
same,  and  at  the  same  time  the  differential  movement  continues 
to  be  a  certain  arc  of  the  orbit,  the  wave-length  remains  the 
same,  but  its  height  and  steepness  increase  (compare  Figs,  i 
and  3).  If  the  size  of  the  orbit  be  increased  and  the  differential 
movement  remain  the  same  in  absolute  amount,  instead  of  the 
same  in  arc,  the  shape  of  the  wave  will  be  preserved  and  its 
dimensions  increased  with  the  dimensions  of  the  orbit  (compare 
Figs.  2  and  3).  If  the  differential  movement  exceeds  a  certain 
limit  the  curve  will  loop  (see  Fig.  4).  This  condition  corre- 
sponds to  that  of  breaking  waves  as  noted  above. 

Movement  of  particles  below  the  surface. —  If  a  series  of  equidis- 
tant particles  be  considered  which  lie  in  a  vertical  line  in  still 
water,  the  movement  of  the  topmost  or  surface  particle  is  repre- 
sented by  any  one  of  the  orbits  considered  above.  That  of  the 
second  one  is  similar  in  every  way  except  in  size  of  orbit  and 
hence  in  velocity.     Its  orbit  is  smaller  and  described  in  the  same 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE         5 

time.  The  two  particles  have  always  the  same  phase  and  hence 
their  movements  are  parallel  at  a  given  instant.  The  same  is 
true  of  the  third  particle  and  all  below  it,  the  orbits  decreasing 
in  a  descending  geometrical  progression  (Fig.  7).' 

This  fact  is  to  be  taken  with  one  above  stated,  namely,  that 
if  orbits  be  decreased  while  the  angular  differential  movement 
remains  constant,  the  sharpness  of  the  trochoid  curve  is  reduced. 
It  results  from  these  properties  that  in  a  breaker  where  the  curve 
of  the  surface  would  intersect  itself,  and  is  therefore  impossible, 
the  trochoids  below  the  surface  would  show  less  of  looping 
until  a  level  is  reached  where  normal  wave  motion  is  going  on 
(compare  Figs.  4,  5  and  6). 

Lines  of  like  phase. —  If  the  orbits  of  a  vertical  series  of  par- 
ticles be  represented  in  diagram  (see  Fig.  7)  and  the  correspond- 
ing points  on  the  circles  be  connected  with  lines,  then  the  line 
connecting  the  highest  points  and  that  connecting  the  lowest 
points  of  the  several  orbits  are  seen  to  be  straight  and  vertical. 
The  remaining  lines  are  curved  and  inclined.  In  Fig.  8  these 
lines  of  like  phase  are  shown  in  the  positions  where  they  occur 
in  the  wave.  The  particles  ranged  along  any  one  of  these  lines 
would  be  in  a  vertical  line  if  the  water  were  at  rest,  just  as  all 
particles  on  one  of  the  trochoid  curves  would  lie  in  a  horizontal 
line.* 

Consequences  of  the  trochoidal  form  and  of  decreasing  orbits 
below. —  If  a  horizontal  plane  be  passed  midway  between  the 
level  of  the  crests  and  that  of  the  troughs  it  will  pass  through 
the  centers  of  the  orbits  described  by  the  surface  particles.  All 
the  water  at  the  surface  above  this  plane  will  then  have  a  for- 
ward component,  and  all  the  water  at  the  surface  below  this  plane 
will  have  a  backward  component.  An  inspection  of  the  dia- 
grams will  show   that   the   crests   are   steeper  and   shorter  than 

^Rankine,  loc.  cit.  p.  131.  Russell,  "  Report  on  Waves  made  to  the  meeting 
of  the  British  Association  1842-3,"  reprinted  in  The  Wave  of  Translation,  London, 
1885,  gives  formulae  adapted  from  Gerstner  for  the  rate  of  orbitical  diminution  with 
depth. 

»RANKINE,/<7f.  «V.,  p.   129. 


N,  M,  FENNEMAN 


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PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE         7 

the  troughs.  This  contrast  increases  as  the  wave  shortens  (com- 
pare Figs.  I,  2,  and  5).  The  crests  have  not. a  sufficient  volume 
of  water  to  fill  the  troughs,  and  hence  the  level  of  the  water 
at  rest  is  lower  than  the  level  of  the  centers  of  the  orbits  which 
the  surface  particles  describe.  The  lifting  of  the  mean  position 
of  particles  above  their  normal  level  gives  a  store  of  potential 
energy  in  the  wave  in  addition  to  the  kinetic  energy  of  the 
motion  of  the  water.  It  may  be  shown  that  this  energy  of 
position  is  exactly  equal  in  amount  to  the  energy  of  the  water's 
motion.'  This  lifting  and  this  store  of  potential  energy  are  at 
a  maximum  when  the  wave  has  its  greatest  steepness  (when  it 
has  the  cycloid  form). 

The  surface  layer  is  thus  divided  into  strips,  in  one-half  of 
which  the  water  is  moving  forward,  while  in  the  other  half  it  is 
moving  backward  at  the  same  rate.  The  peculiarity  of  the  case 
lies  in  the  fact  that  the  backward  moving  strips  are  broader  than 
those  moving  forward.  Fig.  8  shows  that  the  same  is  true  in 
less  degree  of  layers  below  the  surface.  Nevertheless,  the 
amounts  of  water  moving  in  the  two  directions  are  equal  because 
of  the  greater  thickness  of  the  layers  in  the  forward  moving 
strips.  The  contrast  of  thickness  and  also  of  breadth  disappears 
with  depth. 

MOVEMENTS    DURING   WIND. 

Effect  on  size  and  form. — The  immediate  effect  of  wind  in  the 
direction  of  wave  movement  is  to  accelerate  the  movement  of 
the  particles  on  the  crest.  It  also  retards  the  backward  motion 
of  those  in  the  trough,  but  this  effect  is  smaller  because  these 
particles  are  largely  protected  from  the  wind.  The  result  is 
(i)  an  increase  in  the  size  of  the  orbits;  (2)  an  increase  in  the 
differential  movement  of  the  particles  at  the  surface;  (3)  more 
rapid  traveling  of  crest  than  trough,  hence  greater  steepness  in 
front.  The  first  would  result  in  increasing  both  height  and 
length  of  wave  in  the  same  proportion.  The  second  results  in 
greater  steepness,  that  is,  a  shortening  in  proportion  to  their 
height.     The  increased  differential  movement  is  accompanied  by 

^Ibid.,  p.  132. 


8  N.  M.  FENNEMAN 

increased  friction  which  comes  at  length  to  consume  all  the 
energy  derived  from  the  wind  which  cannot  then  further  increase 
the  height  of  the  waves.  The  opposite  effects  are  seen  when  the 
wind  has  ceased.  Friction  gradually  diminishes  the  differential 
movement  of  particles  and  the  size  of  the  orbits.  Waves  then 
become  lower  and  at  the  same  time  longer  in  proportion  to  their 
height. 

Periodical  large  waves, — The  change  of  wave-length  must  be 
propagated  downward  gradually.  If  such  propagation  were 
immediate,  the  wave-length  at  the  surface  would  always  be 
equal  to  that  below.  Not  being  immediate,  there  may  be  at 
times  considerable  differences  in  length.  The  periodical  large 
waves  always  seen  in  a  storm,  may  result  from  composition  of 
lower  and  upper  waves  having  different  periods,  as  well  as  by 
composition  of  surface  waves  of  different  systems. 

WAVES    IN    WATER    OF    FINITE    DEPTH. 

Wave  base. — The  extent  of  orbital  movement  decreases  in 
geometrical  progression  with  depth.  A  point  is  therefore  reached 
where  the  force  is  too  small  to  overcome  the  viscosity  of  the 
water.  Before  this  point  is  reached  and  at  comparatively  small 
depths  the  movement  is  so  slight  that  it  cannot  affect  the  small- 
est solid  particles  resting  on  the  bottom.  This  level,  below 
which  the  largest  waves  are  inoperative,  has  been  called  wave- 
base.  Its  depth  for  any  given  lake  or  part  of  the  ocean  is  a  func- 
tion of  the  height  and  length  of  the  largest  waves. 

Behavior  of  water  above  wave-base  in  pure  oscillation. —  Before 
considering  the  action  of  water  on  a  bottom  which  lies  above 
wave-base  it  will  be  convenient  to  examine  its  behavior  at  any 
horizontal  plane  passed  through  a  system  of  waves.  Referring 
to  Fig.  9,  let  AB  be  such  an  ideal  plane.  Being  above  wave- 
base  it  is  in  the  region  where  the  "planes  of  continuity"  (planes 
including  always  the  same  particles  which  are  in  a  horizontal 
plane  when  at  rest)  are  in  trochoid  curves.  The  lines  of  like 
phase  are  inclined  toward  the  crests ;  hence  the  layer  of  water 
included  between  two  planes  of  continuity  is  not  onl})  thinner 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE         9 

but  broader  under  the  troughs  than  under  the  crests.  In  any  one 
such  layer  the  water  is  moving  forward  under  the  crests  (point  D, 
Fig.  9)  at  the  same  velocity  with  which  it  moves  backward 
under  the  troughs  (point  E  of  same  layer) .  Of  two  adjacent 
layers  the  lower  one  is  composed  of  slower-moving  water.  The 
line  AB,  drawn  in  a  horizontal  plane,  traverses  higher  layers  of 
water  under  troughs  (at  point  C)  and  lower  layers  under  crests 
(point  D).  Therefore  the  backward  moving  water  along  this 
plane  has  a  more  rapid  motion  than  that  moving  forward.  The 
area  covered  by  it  on  the  horizontal  plane  is  also  more  than  that 
covered  by  the  forward  moving  water.  This  excess  of  backward 
movement  below  is  the  necessary  correlative  of  the  excess  of 
forward  movement  above,  for  above  the  plane  traversing  the 
centers  of  the  topmost  orbits  the  movement  on  all  planes  is 
forward. 

The  same  when  waves  are  wind-driven. —  If  now  the  water  be 
conceived  to  be  driven  by  a  wind,  the  current  movement  pro- 
duced at  any  given  depth  must  be  added  to  the  forward  move- 
ment in  the  corresponding  strips  which  lie  below  the  crests,  and 
subtracted  from  the  backward  movement  in  those  under  the 
troughs.  The  forward  and  backward  velocities  in  any  one 
layer  between  two  trochoidal  planes  are  now  no  longer  equal. 
When  a  certain  rate  of  current  is  reached,  the  forward  move- 
ment in  the  lower  layer  traversed  by  the  horizontal  plane  under 
the  crests  (point  D,  Fig.  9)  will  equal  the  backward  movement 
in  the  upper  layer  which  the  plane  traverses  under  the  troughs 
(point  C).  A  certain  force  of  wind  will  therefore  cause  a  bal- 
ance of  to-and-fro  movements  at  a  horizontal  plane  below  the 
surface.  Any  greater  force  will  cause  an  excess  of  forward 
motion. 

Film  representing  surface  of  continuity, —  If  in  one  of  the  sur- 
faces of  continuity  in  a  system  of  waves  of  pure  oscillation,  a 
film  could  be  introduced  which  is  perfectly  flexible  and  friction- 
less,  this  film  would  show  alternate  depressions  and  elevations 
corresponding  to  those  on  the  surface  of  the  water,  but  less 
sharply  curved.     The  curves  would  progress  after  the  manner  of 


10  N.  M.  FENNEMAN 

surface  waves.  Any  one  point  in  the  film  would  rise  and  fall 
vertically  ;  any  particle  of  water  adjacent  to  it  would  continue 
to  describe  its  normal  circle,  gliding  to-and-fro  on  the  friction- 
less  film  and  tracing  a  straight  line  upon  its  surface.  The 
diameter  of  this  orbit  is  represented  by  the  vertical  distance 
through  which  any  point  in  the  film  swings.  If  the  water  above 
the  film  be  viewed  in  cross  section,  the  area  in  which  it  is  mov- 
ing forward  would  equal  that  in  which  it  moves  backward. 

Action  on  a  solid  horizontal  plane  surface. —  If  the  film  be  sup- 
posed now  to  be  stretched  to  a  horizontal  plane  and  to  become 
a  solid  bottom  of  the  ordinary  kind,  several  changes  become 
necessary  in  the  behavior  of  the  adjacent  water  particles.  The 
up-and-down  movement  in  their  orbits  becomes  impossible,  but 
the  to-and-fro  movement,  tracing  straight  lines  on  the  surface, 
can  be  continued.  Observation  shows  that  this  does  occur,  that 
particles  near  a  shallow  bottom  move  back  and  forward  in  straight 
lines,  and  that  vertical  movement  gradually  appears  in  the  paths 
of  higher  particles,  these  paths  being  at  first  very  flat  ovals,  but 
becoming  higher  and  more  nearly  circular  as  the  surface  is 
approached.* 

The  energy  of  the  vertical  movement  thus  interfered  with  is 
partly  expended  in  friction  on  the  bottom,  though  it  is  quite 
possible  that  a  part  of  it  may  be  used  in  an  increased  horizontal 
amplitude.''  It  is  a  matter  of  observation  that  this  flattening  of 
orbits  affects  the  movements  of  surface  particles  as  well  as  of 
those  below. 3  This  effect  on  the  topmost  orbits  is  in  proportion 
to  the  degree  of  interference  at  the  bottom.  Very  much  elon- 
gated orbits  indicate  large  friction,  just  as  circular  orbits  indi- 
cate that  there  is  no  appreciable  interference  at  the  bottom. 
^  Effect  on  wave-length,  etc. —  The  immediate  effect  of  retarda- 
tnn  of  particles  in  contact  with  the  bottom  must  be  an  increased 

»  Weber,  Wellenlehre,  p.  124. 

*  The  observations  of  the  brothers  Weber,  as  recorded  in  the  table  given  in  Wel- 
lenlehre,^.  124,  seem  to  show  that  the  horizontal  motion  on  a  shallow  bottom,  while 
less  than  at  the  surface,  is  actually  greater  than  a  certain  intermediate  point. 

3  Weber,  loc.  cit. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE       II 

differential  movement  of  adjacent  water  particles.  The  laws  of 
fluids  require  that  this  differential  movement  be  distributed 
throughout  the  series  of  particles  reaching  to  the  surface,  though 
experienced  to  a  less  degree  as  the  distance  from  the  bottom 
increases.  It  has  been  shown  that  increased  differential  move- 
ment implies  decreased  wave-length.  This  shortening,  accom- 
panied by  steepening,  may  or  may  not  be  sufficient  to  cause 
breaking.  Since  these  effects  are  greater  at  the  bottom  than  at 
the  surface,  the  lines  of  like  phase  will  incline  forward.  These 
effects  —  the  increased  differential  movement,  the  shortening 
waves,  and  the  forward  inclination  of  lines  of  like  phase — fol- 
low from  friction  on  the  bottom,  but  it  is  immaterial  whether 
this  friction  be  that  of  the  forward-moving  or  that  of  the  back- 
ward-moving water.  The  forward  inclination  of  lines  of  like 
phase  indicates  nothing  as  to  the  movement  of  the  water  as  a 
body.  The  inclination  of  these  lines  may  be  arrived  at  in 
another  way.  The  retarded  particles  below  may  be  thought  of 
as  having  a  decreased  angular  velocity,  and  hence  a  less  advanced 
phase  than  the  upper  particles  in  the  same  vertical  line.  This 
would  require  that  lines  of  like  phase  should  connect  them  with 
upper  particles  in  advance  of  them  in  the  direction  of  wave 
movement. 

Comparison  of  friction  in  forward  and  backward  movement. — 
Looked  at  in  cross-section,  the  area  of  the  backward-moving 
water  above  the  line  AB  (Fig.  9)  is  less  than  the  area  of  for- 
ward-moving water.  The  areas  are  equal  when  bounded  below 
by  one  of  the  trochoid  curves.  The  area  of  backward-moving 
water  is  made  smaller  by  the  substitution  of  a  rigid  plane  for  the 
depressed  part  of  the  trochoid,  and  that  of  the  forward-moving 
water  is  made  larger  by  the  substitution  of  a  flat  bottom  for  the 
curve  bulging  upward.  This  constriction  and  consequent  greater 
friction  of  the  backward-moving  water  makes  itself  felt  in  the 
form  of  the  wave  and  in  the  bodily  movement  of  the  water. 

Asymmetrical  form. —  The  velocity  of  propagation  of  wave 
crests  depends  purely  upon  the  behavior  of  particles  in  the 
upper  halves  of  their  orbits,  while  the   propagation   of  troughs 


12  N.  M.  FENNEMAN 

takes  account  of  the  lower  halves  only.  It  results  from  a 
greater  orbital  velocity  in  the  upper  halves  that  crests  are  propa- 
gated more  rapidly  than  troughs.'  The  necessary  accompani- 
ment of  this  is  the  asymmetrical  form,  steeper  in  front  than 
behind. 

Resulting  currents. — The  constriction  of  backward-moving 
water  mentioned  above  may  be  compensated  either  by  greater 
velocity  or  by  broadening  the  area  of  backward  flow.  Upon 
either  of  these  assumptions,  or  upon  the  assumption  of  no  com- 
pensation, certain  conclusions  follow  from  a  geometrical  inspec- 
tion of  the  diagram,  and  these  conclusions  agree  with  observed 
phenomena. 

Assume  first  that  the  deficiency  in  backward  movement  is 
uncompensated.  This  assumption  involves  an  excess  of  forward 
movement  which  would  be  observed  as  a  current,  a  well-known 
phenomenon  where  waves  enter  shallow  water.  On  this  same 
supposition  of  no  compensation  the  area  of  the  bottom  covered 
by  the  backward-moving  water  is  greater  than  that  covered  by 
the  forward-moving  water,  and  the  velocity  of  that  moving  back- 
ward on  the  bottom  is  greater  than  of  that  moving  forward  at 
the  same  depth,  because  the  former,  being  under  the  trough,  is 
nearer  to  the  surface.  A  current  of  this  type  would  therefore  be 
distinctly  a  surface  feature  which  would  not  wash  the  bottom  in 
the  direction  of  its  flow.  It  would,  in  fact,  involve  a  certain 
amount  of  counter-current  at  the  bottom,  independent  of  any  of 
the  conditions  which  give  rise  to  undertow. 

Assume  next  that  the  deficiency  of  area  of  backward  mov- 
ing water  is  compensated  in  one  of  the  ways  above  mentioned, 
either  by  greater  velocity  or  by  broadening  the  area.  In  either 
of  these  cases  the  backward  movement  on  the  bottom  will  be  in 
excess,  and  will  suffer  more  interference  by  friction  than  the 
forward  movement  will.  This  greater  interference  with  the 
backward  movement  will  favor,  with  each  oscillation,  a  residual 
advance  of  the  water  as  a  whole,  causing  a  progression  in  the 

*  Compare  also  C.  S.  Lyman,  "A  New  Form  of  Wave  A.^i^^xz.ins,,'"  Journal  of  the 
Franklin  Institute,  Vol.  LXXXVI,  p.  187. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE       1 3 

direction  of  wave  movement,  by  a  process  which  has  something 
in  common  with  walking.  In  this  way  also,  pure  oscillation 
would  give  rise  to  a  current. 

It  is  evident  then,  that  when  a  system  of  waves  of  pure 
oscillation  advances  over  a  shallow  bottom,  any  supposition  that 
may  be  made  regarding  the  adjustment  of  internal  movements 
will  result  in  a  forward  flow  of  water  above,  and  a  dominance  of 
movement  in  the  opposite  direction  below.  Owing  to  friction, 
the  latter  alone  is  never  equal  to  the  former.  The  resulting 
movement  of  water  in  the  direction  of  wave  propagation,  whether 
it  be  viewed  as  a  current  or  as  an  increase  of  the  positive  over 
the  negative  parts  of  ordinary  waves,  is  not  the  same  as  waves 
of  translation,  technically  so  called.^  These  latter  obey  differ- 
ent laws  and  move  with  different  velocities.  They  may  be 
occasioned  by  breakers,  or  may  perhaps  grow  out  of  oscillatory 
waves  by  gradual  transition,  but  their  movements  are  character- 
ized by  certain  features  to  be  mentioned  later. 

The  return  current. — As  soon  as  a  current  is  initiated  a  return 
of  the  water  becomes  necessary.  If  the  process  described  above 
be  supposed  to  take  place  on  a  shoal  without  shores  this  return 
may  take  place  by  another  route.  In  this  case  the  current  may 
proceed  as  described  for  an  indefinite  time.  If  there  is  no 
return  over  another  area  by  horizontal  circulation,  then  the  return 
must  be  over  the  same  area  by  vertical  circulation;  that  is,  either 
above  or  below  the  original  current.  If  the  forward  orbital 
movement  above  exceed  the  backward  orbital  movement  below, 
as  seems  necessary,  and  no  lateral  escape  is  at  hand,  the  pressure 
due  to  increased  height  of  the  water  would  cause  a  counter  cur- 
rent which  would  appear  below  as  undertow. 

Action  on  bottom  materials. — The  essential  value  of  the  consid- 
eration of  these  currents,  springing  from  waves  of  pure  oscilla- 
tion, is  in  the  necessary  conclusion  that  the  work  of  such  waves 
is  backward  at  the  bottom,  and  7iot  forward.  The  advance  of  the 
water  described  is  due  to  interference  with  its  backward  flow. 
The  same  friction  which  impedes  the  backward  movement  of  the 

*  J.  Scott  Russell,  The  Wave  of  Translation. 


14  N.  M.  FENNEMAN 

water  causes  the  motion  which  the  water  loses  to  be  communi- 
cated to  the  materials  of  the  bottom.  The  case  is  roughly 
analogous  to  the  wheels  of  a  locomotive,  which  in  "flying  the 
track"  brush  the  sand  on  the  track  backward. 

The  case  of  wind- driven  waves, — ^The  above  case  is  applicable 
only  to  waves  of  pure  oscillation,  which  have  of  necessity  been 
generated  in  deep  water  and  are  advancing  over  a  shallow 
bottom.  If  the  wind  is  blowing  at  the  same  time  in  the  direc- 
tion of  wave  movement,  the  result  will  be  similar  to  that  found 
in  considering  a  mathematical  plane  above  wave-base,  provided, 
of  course,  that  the  return  of  the  water  is  by  horizontal  circula- 
tion. The  action  of  the  wind  increases  the  forward  motion 
under  crests  and  diminishes  the  backward  motion  under  troughs. 
When  the  effect  of  this  action  reaches  a  certain  amount,  the 
influences  named  above,  which  give  dominance  to  the  backward 
movement  at  the  bottom,  will  be  counterbalanced,  and  any 
greater  effect  of  the  wind  will  give,  at  the  bottom,  an  excess  of 
forward  movement.  A  moderate  effect  of  the  wind  is  probably 
usually  sufficient  to  overcome  the  backward  brushing  due  to 
oscillation  alone.  If  the  return  is  by  vertical  circulation,  any 
increase  in  current  above  involves  an  increased  reverse  current 
below. 

The  case  of  breaking  waves. — When  waves  generated  in  deep 
water  advance  over  a  bottom  sufficiently  shallow  to  cause 
breaking,  a  new  factor  is  introduced.  In  this  case  there  is  a 
tendency  to  the  formation  of  positive  waves  of  translation, 
which  may  sometimes  develop  typically,  though  doubtless  more 
often  their  motion  enters  in  merely  as  a  component.  It  is  in 
the  nature  of  these  that  all  the  particles  in  and  under  the  wave 
form  move  forward  and  not  backward,  and  the  forward  motion 
is  the  same  at  all  depths.'  To  the  extent  that  this  factor  enters, 
the  effect  on  the  bottom  will  of  course  be  to  urge  material  in 
the  direction  of  wave  movement. 

» See  Russell,  The  Wave  of  Translation,  p.  42 ;  Report  on  Waves,  p.  307 ;  also 
D'AURIA,  "A  New  Theory  of  the  Propagation  of  Waves  in  Liquids,"  yi?«r«a/  of  the 
Franklin  Institute,  1890,  p.  460.     The  last  named  is  a  mathematical  discussion. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE       1$ 
WAVES    IN    SHALLOWING    WATER. 

Tendency  to  enlargement  of  orbits. — When  a  system  of  waves 
generated  in  deep  water  reaches  shallow  water,  certain  forces 
operate  to  increase  the  sizes  of  the  orbits,  while  others  produce 
the  opposite  tendency.  In  general  the  increase  of  orbital 
motion  is  due  to  the  transmittal  of  the  motion  of  a  larger 
amount  of  water  to  a  smaller  amount.^ 

If  the  shallow  water  be  separated  from  the  deep  water  by  a 
vertical  face  i^BC  in  Fig.  lo),  the  change  may  operate  in  some 
manner  similar  to  the  following :  The  deep  water  on  the  right 
side  of  the  figure  is  agitated  to  the  depth  of  C  by  waves  travel- 
ing toward  the  left.  The  motion  of  particles  below  the  level  of 
B  is  influenced  by  the  vertical  face  BC,  this  influence  being 
greater  in  proportion  to  their  nearness.  Those  in  contact  with 
the  surface  must  move  in  straight  lines  up  and  down,  while 
those  farther  away  describe  ovals  whose  longer  diameters  are 
vertical,  and  whose  shapes  become  more  circular  with  distance 
from  BC.  The  energy  of  the  horizontal  motion  thus  lost  is,  of 
necessity,  partly  expended  in  friction  on  the  vertical  face.  That 
which  remains  must  be  devoted  to  increasing  the  vertical  move- 
ment. By  this  means  it  is  again  communicated  to  the  particles 
above  the  level  of  B. 

If  the  change  from  deep  to  shallow  water  be  gradual,  the 
analysis  of  the  process  is  essentially  the  same.  In  this  case^ 
however,  the  circular  orbits  below  will  give  way  to  straight  line 
movement,  not  vertical,  but  parallel  to  the  sloping  bottom  DC. 
As  before,  friction  will  consume  a  part  of  the  energy  which 
orbital  motion  has  lost,  the  remainder  being  expended  in 
increased  movement  parallel  to  the  sloping  bottom.  Of  this 
movement  the  vertical  component  will  go  to  increasing  the  ver- 
tical axis  of  the  orbits  above. 

Tendency  to  diminishing  orbits, — Along  with  the  above  tendency 
to  increased  orbits  come  two  tendencies  toward  diminution. 
The  first  of  these  is  the  influence  of  the  flatter  orbits  of  the 
lower  particles.     It   tends   to   diminish   the  vertical  movement 

*  Compare  C.  S.  Lyman,  loc.  cit.,  p.  193. 


l6  N.  M.  FENNEMAN 

above,  but  not  the  horizontal.  The  second  influence  toward 
diminution  is  the  friction  on  the  bottom  which  is  shared  by  the 
particles  above. 

Opposite  tendencies  simultaneous. —  On  a  sloping  surface  the 
opposing  tendencies  act  at  the  same  time.  It  is  evident  that  in 
proportion  as  the  slope  is  steep,  sudden  enlargement  will  be 
favored,  and  that  slow  shallowing  favors  reduction  in  size 
because  of  the  long  continued  action  of  friction.  Theoretically, 
there  should  be  a  grade  on  which  an  incoming  wave  should 
suffer  no  change  of  height,  but  since  the  form  and  internal 
movements  would  change,  this  ideal  grade  is  not  of  importance 
in  considering  the  work  of  water  on  the  bottom. 

Tendency  to  decreased  wave-length. —  If  the  supposed  tendency 
toward  orbital  increase  be  balanced  by  the  opposite  tendency 
arising  from  friction,  there  will,  of  course,  be  no  increase  in  the 
length  or  height.  However,  when  waves  do  increase  in  height, 
showing  that  the  orbits  have  enlarged,  they  are  still  very  com- 
monly diminished  in  length  and  of  necessity  increased  in  steep- 
ness. This  is  readily  explained  by  the  increased  differential 
movement  of  particles,  initiated  by  friction  on  the  bottom. 

TendeTicy  to  steepening  due  to  wind. — The  largest  on-shore 
waves  usually  act  in  conjunction  with  the  wind  blowing  in  the 
approximate  direction  of  their  movement.  The  effect  of  wind 
on  waves  in  deep  water  was  seen  to  be  similar  to  the  effect  of  a 
shallow  bottom,  namely,  (i)  increase  of  orbits;  (2)  increase  of 
steepness  ;  (3)  asymmetrical  form.  These  effects  may  be  carried 
to  the  point  of  breaking,  even  in  water  of  infinite  depth  (white- 
caps).  On  a  shallow  bottom  the  effects  are  increased  by  the 
concurrent  action  of  the  two  factors.  Where  there  is  no  wind 
waves  are  commonly  supposed  to  break  in  water  whose  depth  is 
equal  to  or  a  little  greater  than  the  height  of  the  waves  above 
the  level  of  repose.*  When  waves  advancing  on  a  shallow  bot- 
tom are  already  strained  by  the  wind,  they  may  break  with  much 
regularity  in  much  greater  depths  of  water,  equal  to  perhaps  two, 
three,  or  four  times  the  height  of   the  wave.     Thus  while  the 

'Russell,  Report  on  Waves,  p.  245. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE       I J 

breaker  line  for  waves  without  wind  is  far  up  the  slope  from 
wave-base,  it  may  move  down  indefinitely  near  to  wave-base 
when  the  wind  is  active. 

Tendeficy  of  wave  to  recover  form. —  Suppose  a  system  of 
oscillatory  waves  to  advance  toward  a  shelving  shore  until  the 
wave-base  intersects  the  bottom.  One  effect  must  be  produced 
here  regardless  of  qualifying  conditions.  Bottom  friction  begins 
and  that  involves  increased  differential  movement  of  particles, 
which  is  accompanied  by  shortening  and  steepening  of  waves. 
This  implies  increased  internal  friction,  which  in  turn,  operates 
to  decrease  the  orbital  motion  and  therefore  wave  dimensions. 
In  so  doing  it  would  take  away  the  conditions  of  bottom  friction 
and  its  results.  The  wave  would  then  return  to  its  deep  water 
form.  Thus  there  is  a  chain  of  consequences  from  the  original 
interference  at  the  bottom,  which  involves  at  first  the  change  of 
wave  form,  but  later  a  restoration,  the  final  result  being  reduc- 
tion in  dimensions  only,  suited  to  the  diminished  depth.  Another 
decrease  of  depth  must  then  be  assumed  if  the  wave  be  supposed 
to  continue  its  contact  with  the  bottom.  Thus  there  is  a  certain 
minimum  slope  for  the  bottom,  upo?i  which  the  waves  may  be  propa- 
gated as  a  shallow-water  wave.  In  so  far  as  the  wave  is  affected 
by  increase  of  orbit  due  to  diminishing  amount  of  water,  the 
effect  will  be  to  hasten  the  deformation  and  to  retard  the  recov- 
ery of  form.  If  the  wind  is  active  it  would  retard  the  decrease 
of  orbital  movement  and  the  minimum  slope  mentioned  would 
be  smaller. 

Limit  of  tendency  to  recover  form. — The  greater  the  reduc- 
tion of  depth,  the  greater  the  increment  of  internal  friction 
tending  to  reduce  the  wave  size,  and  the  greater  this  friction,  the 
more  rapidly  does  it  operate  to  accommodate  the  wave  dimensions 
to  diminished  depth.  This  corrective  tendency  has,  however,  a 
limit.  This  limit  is  marked  by  the  breaking  of  the  wave.  There 
is,  therefore,  a  certain  maximum  slope  for  the  bottom  upon  which  the 
wave  m,ay  be  propagated  without  breaking;  at  or  beyond  this  maxi- 
mum the  wave  breaks  and  other  agencies  come  in.  The  effect 
of  wind  as  before,  is  to  diminish  the  maximum  slope ;   hence 


1 8  N.  M.  FENNEMAN 

true  breakers  (not  whitecaps  merely)  may  occur  during  a  wind 
on  a  shore  where  waves  of  the  same  size  would  not  break  in  a 
calm. 

Effect  of  breaking  on  wave  propagation. —  Even  when  the 
distortion  of  wave  form  has  been  pressed  beyond  the  breaking 
point,  the  effort  to  recover  its  form  and  habit  does  not  cease. 
This  effort  is  now  favored  by  all  the  tendencies  which  existed 
before  breaking  and  re-enforced  by  one  more  arising  from  the 
falling  crests.  As  shown  in  the  diagram  (Fig.  4),  breaking  is 
an  expression  of  conflicting  orbits.  The  water  above  the  node 
of  the  hypothetical  surface  does  not  continue  the  curve  which  it 
has  been  describing,  but  falls  confusedly  on  the  front  of  the 
wave.  Here  its  downward  motion  is  in  direct  opposition  to  the 
upward  motion  of  the  water  in  front  of  the  crest.  Thus,  to  the 
molecular  resistance  of  friction,  is  added  mass  conflict,  both  of 
which  operate  to  reduce  wave  motion.  This  reduction  is  there- 
fore accomplished  more  rapidly  than  in  the  case  of  unbroken 
waves.  It  results  from  this,  that  waves  often  break  at  some  dis- 
tance from  shore,  and  after  traveling  a  short  distance  with  foam- 
ing crests,  recover  their  form  and  advance  a  long  distance  with 
crests  entire.  There  is  a  certain  slope  on  which  waves  will 
advance  with  nearly  uniform  shape  and  continuously  foaming 
crests.  On  a  gentler  slope  they  will  recover  their  unbroken 
form  ;  on  a  steeper  slope  the  first  breaking  occurs  close  to  shore, 
and  the  wave  form  is  speedily  lost. 

Waves  of  translation. — When  waves  of  oscillation  enter  shallow 
water  the  habit  of  the  water  particles  changes  and  becomes  a 
compromise  between  orbital  oscillation  and  movement  of  an 
entirely  different   nature,   belonging  to   waves   of    translation.' 

The  essential  features  of  the  positive  wave  of  translation, 
known  also  as  the  wave  of  the  first  order  or  the  solitary  wave  are, 
( I )  it  is  initiated  by  an  elevation  of  the  water  surface  above  its 
normal  level;  (2)  it  is  propagated  without  a  corresponding 
trough  and  without  companion  crests,  being  entirely  above  the 
undisturbed  level  of   repose;    (3)   its  rate  of   travel  is  greater 

*  Russell,  Report  on  Waves  and  Wave  of  Translation. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE       \^ 

than  that  of  waves  of  oscillation,  when  like  wave-lengths  are 
assumed,  the  two  rates  having  about  the  ratio  of  three  to  two;* 
(4)  the  water  particles  move  forward  and  not  backward,  start- 
ing from  rest  as  the  wave  approaches  and  coming  to  rest  when- 
the  wave  has  passed ;  (5)  the  forward  motion  of  particles  at  all- 
depths  is  the  same  and  equal  to  the  volume  of  the  wave  divided 
by  the  depth  of  the  water;  (6)  the  paths  of  the  particles  are: 
semi-ellipses  in  a  vertical  plane,  the  major  axis  being  the  dis- 
tance through  which  the  particle  moves  forward,  and  the  minor 
axis  varying  from  zero  at  the  bottom  to  the  height  of  the  wave 
at  the  surface.  This  movement  is  in  no  sense  the  same  as  that 
of  wind-driven  waves  or  any  other  oscillatory  wave  motion  com- 
pounded with  current.  It  usually  coexists  with  the  latter  on 
shallow  bottoms,  resulting  in  waves  of  a  hybrid  kind ;  but  waves 
of  nearly  typical  translatory  character  may  sometimes  be  seen  in 
nature.  Whether  the  waves  be  of  a  pure  or  mixed  type,  the 
essential  fact  here  is  that  a  nt.^  factor  has  entered,  whose  action 
at  the  bottom  is  different  from  that  of  oscillatory  waves  and 
from  that  of  currents. 

The  fact  of  this  change  to  translatory  character  on  a  gently 
sloping  beach  may  be  seen  in  the  behavior  of  floating  chips 
which  are  seen  to  move  forward  on  crests  but  not  backward 
between  crests.  In  place  of  the  trough  proper  is  a  wide  strip 
whose  surface  is  almost  flat  and  the  water  of  which  is  standing 
still.  The  laws  of  translatory  waves  require  that  they  move 
more  rapidly  than  the  oscillacory.  This  might  be  expected  to 
reveal  itself  in  broadening  intervals  between  crests  as  waves  take 
on  the  translatory  character.  It  is  probable  that  this  may  occur 
under  suitable  conditions.  The  tendency  is  usually  more  than 
counterbalanced  by  two  factors.  The  first  is  the  decreasing  depth 
which  is  the  main  factor  in  controlling  the  velocity  of  waves  of 
,  translation.  The  second  is  the  increasing  strength  of  undertow 
near  shore  which  retards  the  translatory  movement  at  the  bot- 
tom. 

As  to  the  manner  in  which  this  new  habit  is  developed,  it  may 

^  Ibid., -p.  288. 


20  N  M.  FENNEMAN 

be  cited  that  perfect  waves  of  the  first  order  are  produced  experi- 
mentally by  the  sudden  addition  of  water  at  one  end  of  a 
rectangular  vessel,  or  by  the  immersion  of  a  solid,  or  by  a 
sudden  pushing  forward  of  the  wall  of  the  vessel,  the  effect  in 
«ach  case  being  the  local  raising  of  the  water  surface  above  the 
level  of  repose.  A  corresponding  process  in  lakes  or  sea  where 
the  bottom  becomes  shallow  may  be  found  in  the  sudden  deliv- 
-ery  of  the  mass  of  water  which  falls  upon  the  front  of  a  breaking 
wave.  Observation  on  the  shores  of  large  water  bodies,  such  as 
the  great  lakes,  would  indicate  that  the  area  over  which  waves 
show  a  translatory  element  is  somewhat  definitely  limited  by  the 
breaker  line.  It  is  probable,  however,  that  there  is  also  a  more 
gradual  change  by  which  the  waves  become  increasingly  positive 
as  the  water  shallows  and  the  features  of  waves  of  the  first  order 
are  thereby  assumed. 

If  the  modifications  of  oscillatory  waves  in  shallowing  water 
be  reviewed  while  holding  in  mind  the  charactertistics  of  trans- 
latory waves  as  given  above,  it  will  be  observed  that  these 
changes  are  all  in  the  direction  which  would  favor  the  conver- 
sion of  oscillatory  into  translatory  waves.  This  is  seen  in  the 
increase  of  crests  with  corresponding  disappearance  of  troughs  ; 
the  growing  excess  of  the  forward  movement  of  particles  over 
backward  movement  and  the  increased  horizontal  amplitude  of 
the  lower  orbits,  approaching  equality  with  that  of  the  orbits 
above.  For  present  purposes  it  may  suffice  to  adopt  the  con- 
ception of  Mr.  Russell^  who  thought  of  the  overgrown  crest  as 

*  The  wave  of  the  second  order  may  disappear  and  a  wave  of  the  first  order  take 
its  place.  The  conditions  under  which  I  have  observed  this  phenomenon  are  as  fol- 
lows :  one  of  the  common  sea  waves,  being  of  the  second  order,  approaches  the  shore, 
consisting  as  usual  of  a  negative  or  hollow  part  and  of  a  positive  part  elevated  above 
the  level ;  and  as  formerly  noted,  this  positive  portion  gradually  increases  in  height 
and  at  length  the  wave  breaks,  and  the  positive  part  of  the  wave  falls  forward  into 
the  negative  part,  filling  up  the  hollow.  Now  we  readily  enough  conceive  that  if  the 
positive  and  negative  parts  of  the  wave  were  precisely  equal  in  height,  volume,  and 
velocity,  they  would  by  uniting,  exactly  neutralize  each  other's  motion,  and  the  volume 
of  the  one,  falling  into  the  hollow  of  the  other,  give  rise  to  smooth  water ;  but  in 
approaching  the  shore  the  positive  part  increases  in  height  and  the  result  of  this  is  to 
Jeave  the  positive  portion  of  the  wave  much  in  excess  above  the  negative.  After  a 
•wave  has  first  been  made  to  break  on  the  shore  it  does  not  cease  to  travel,  but  if  the 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE      21 

falling  forward  into  the  diminished  trough  in  the  act  of  break- 
ing ;  the  trough  is  more  than  filled  and  the  excess  of  water  ini- 
tiates a  wave  of  translation  exactly  as  in  Mr.  Russell's  experi- 
ments. 

Volume  of  undertow . —  It  is  not  necessary  to  suppose  that  the 
loss  of  velocity  of  the  undertow  is  as  rapid  as  the  increase  of  its 
cross-section.  This  would  be  the  case  if  all  the  upper  water 
moving  shoreward  should  reach  the  shore  before  turning  back. 
The  volume  of  the  undertow  would  then  also  be  constant 
throughout  its  course  and  its  velocity  would  be  inversely  as  its 
cross-section.  But  even  if  the  loss  of  motion  due  to  friction  and 
interference  of  the  bottom  be  ignored,  not  all  the  shoreward 
moving  water  reaches  the  shore.  The  on-shore  motion  causes 
elevation  of  level  over  a  belt  of  considerable  width.  This  broad 
elevation  constitutes  a  head  which  is  the  cause  of  outward  flow 
below.  It  may  be  shown  that  the  average  position  at  which 
incoming  particles  turn  back  and  join  the  undertow,  is  at  the 
center  of  mass  of  the  head.  This  head  is  greatest  at  the  edge 
of  the  water,  hence  more  water  turns  back  at  that  point  than  at 
any  other,  but  the  undertow  which  has  its  beginning  here  is 
constantly  being  augmented  by  that  which  returns  toward  deeper 
water  without  reaching  the  shore. 

slope  be  gentle,  the  beach  shallow  and  very  extended  (as  it  sometimes  is  for  a  mile 
inward  from  the  breaking  point,  if  the  wave  be  large)  the  whole  inner  portion  of  the 
beach  is  covered  with  positive  waves  of  the  first  order,  from  among  which  all  waves 
of  the  second  order  have  disappeared.  This  accounts  for  the  phenomenon  of  breakers 
transporting  shingle  and  wreck  and  other  substances  shoreward  after  a  certain  point ; 
at  a  great  distance  from  shore  or  where  the  shores  are  steep  and  abrupt  the  wave  is  of 
the  second  order,  and  a  body  floating  near  the  surface  is  alternately  carried  forward 
and  backward  by  the  waves,  neither  is  the  water  affected  to  a  great  depth ;  whereas, 
near  the  shore  the  whole  action  of  the  wave  is  inwards,  and  the  force  extends  to  the 
bottom  of  the  water  and  stirs  the  shingle  shoreward ;  hence  the  abruptness  also  of  the 
shingle  and  sand  near  the  margin  of  the  shore  where  the  breakers  generally  run. 
....  The  residuary  waves  given  off  after  breaking  are  wide  asunder  from 
each  other,  are  wholly  positive,  and  the  spaces  between  them,  several  times  greater 
than  the  amplitude  of  the  waves,  are  perfectly  flat  and  in  this  condition  they 
extend  over  wide  areas  and  travel  to  great  distances.  These  residuary  positive  waves 
evidently  prove  the  existence,  and  represent  the  amount,  of  the  excess  of  the  positive 
above  the  negative  forces  in  the  wind  wave  of  the  second  order. — Report  on  WaveSy 
p.  292. 


22  N.  M.  FENNEMAN 

Relation  of  the  phenomena  above  to  agitation  on  the  bottom. — 
It  is  to  be  inferred  from  what  precedes  that  symmetrical  wave 
form  indicates  freedom  from  interference  at  the  bottom,  that 
friction  below  is  great  in  proportion  as  crowding,  steepening, 
and  asymmetrical  form  above  are  prominent,  and  that  where  an 
off-shore  breaker  line  is  seen  it  indicates  a  maximum  of  bottom 
interference.  It  is  understood  in  all  cases  that  the  surface  effect 
will  lag  a  little  behind  the  cause  below,  and  therefore  appear  a 
little  to  shoreward. 

PROFILES    RESULTING    FROM    FORCES    DISCUSSED    ABOVE. 

In  the  actual  operation  of  the  forces  discussed  above,  the 
resulting  action  on  a  sloping  bottom  may  be  outward  at  all 
places,  or  inward  at  all  places,  or  outward  over  one  part  and 
inward  over  another.  Forces  in  either  direction  may  be  gradu- 
ually  augmented  or  diminished.  The  different  forces  are  capable 
of  different  combinations.  Each  set  of  conditions  will  lead  to 
certain  features  of  profile,  if  there  be  no  change  of  condition, 
a  permanent  profile  of  equilibrium  may  be  reached.  The  con- 
stant supply  of  load  constitutes  an  ever  shifting  condition. 
Equilibrium  as  commonly  realized  depends  on  the  uniformity  of 
this  supply. 

Factors  in  profile -making. — The  agencies  which  shape  the 
marginal  bottom  may  be  treated  in  three  groups,  ( i )  oscil- 
latory wave  action  and  undertow,  carrying  material  from  shore ; 
(2)  on-shore  currents  and  translatory  wave  action,  carrying  the 
material  toward  the  shore  ;  (3)  currents  alongshore.  The  tend- 
ency of  the  first  group  is  to  steepen  the  slope  from  the  water's 
edge  to  the  line  at  which  its  erosive  power  ceases,  and  deposi- 
tion begins  and  to  reduce  the  slope  beyond  that  line.  There  is 
also  for  the  second  group  a  line  of  maximum  power  on  the 
bottom,  within  which  their  effect  is  to  steepen  the  profile  by 
accumulation  at  the  water's  edge,  and  beyond  which  the  slope 
is  reduced  by  cutting  down.  Currents  alongshore  will  be  intro- 
•duced  later. 

Conflict  between  on-shore  and  ofi-shore  action. — The  first  two 


0 


UNIVFR< 

PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE      23 

pairs  of  agencies  are  in  conflict  as  to  the  direction  in  which 
bottom  materials  are  to  be  moved.  If  all  the  water  which 
moves  shoreward  must  return  over  the  same  area  and  as  a  bot- 
tom current,  this  current  would  seem  to  have  greater  efficiency 
than  the  one  above,  moving  in  the  opposite  direction.  This  is 
certainly  the  case  where  translatory  waves  are  not  favored,  as 
where  the  off-shore  slope  is  steep.  Where  slope  is  gentle  and 
translatory  waves  are  well  developed,  they  have  one  decided 
advantage.  They  are  short  as  compared  with  the  distance  from 
wave  to  wave,  hence  all  the  shoreward  movement  of  the  water 
is  concentrated  into  a  small  portion  of  the  entire  time.  Divers 
are  said  to  feel  the  passing  of  one  of  these  waves  as  a  sudden 
jerk  between  intervals  of  quiet.  The  undertow,  on  the  other 
hand,  has  a  steady  flow  except  as  interrupted  by  these  sudden 
reverses.*  The  laws  of  energy  give  to  these  concentrated  move- 
ments a  much  greater  efficiency  than  to  the  same  amount  of 
motion  more  evenly  distributed  in  time.  On  many  shores  of 
gentle  slope,  sand  is  worked  landward,  and  in  this  process  the 
agency  just  mentioned  is  doubtless  important.  The  effect  here 
referred  to  is  that  of  waves  of  translation  and  is  therefore  inside 
the  breaker  line.  It  might  accumulate  sand  on-shore  but  not 
in  off-shore  barriers.  The  dominance  of  shoreward  action  is 
essentially  temporary  (omitting  currents  alongshore  from  con- 
sideration). Its  effect  is  to  steepen  by  narrowing  the  slope. 
This  steepening,  in  turn,  is  adverse  to  waves  of  translation. 

Laws  of  equilibrium ;  eroding  currents. —  Ignoring  the  presence 
of  a  bank  and  the  load  derived  from  it,  a  current  of  uniform 
power  tends  to  reduce  the  bottom  to  a  level  surface,  that  is,  to 
require  equal  depth  throughout.  Equilibrium  cannot  exist  on  a 
level  bottom  where  the  power  of  the  current  is  unequal  at  dif- 
ferent places.  In  such  cases,  the  depth  must  suffer  a  corre- 
sponding  change   until   the  power   of  water   on   the  bottom    is 

^ Henry  Mitchell,  "On  the  Reclamation  of  Tide-Lands  and  its  Relation  to 
Navigation,"  Report  of  the  U.  S.  Coast  and  Geodetic  Survey,  i86g.  Appendix  5,  p.  85. 
In  tliis  paper  Mr.  Mitchell  takes  the  extreme  view  that  the  sea  restores  to  the  conti- 
nent "  all  the  material  washed  from  its  bluffs  and  headlands."  Certain  exceptions  are 
made  for  islands. 


24  N.  M,  FENNEMAN 

everywhere  the  same.  A  current  of  uniformly  increasing  power 
requires  a  uniformly  increasing  depth,  that  is,  a  plane  slope. 
The  opposite  is  true  for  a  current  of  uniformly  diminishing 
power.  A  current  whose  power  is  augmented  at  an  increasing 
rate^  as,  for  example,  in  geometrical  ratio,  requires  a  descent  to 
deep  water  on  a  curve  which  is  convex  upward.  Increase  of 
power  at  a  diminishing  rate  requires  concavity.  Loss  of  power 
at  increasing  rates,  and  loss  at  diminishing  rates,  require  con- 
cavity and  convexity  respectively. 

Uniform  cuttifig  or  building. —  If  a  uniform  current  on  a  level 
bottom  has  eroding  power,  the  whole  will  be  cut  down  at  the 
same  time,  and  the  bottom  will  remain  level  while  depth  increases. 
In  this  case  the  load  is  furnished  at  all  points  equally,  and  is 
all  carried  forward  at  the  same  rate.  If  load  be  furnished  in 
excess  of  carrying  power,  and  at  all  points  uniformly  (as  from 
top  or  sides),  then  the  level  surface  of  the  bottom  would  be 
preserved  while  depth  would  decrease. 

Load  derived  from  the  shore. — To  make  the  case  applicable  to 
undertow,  the  excessive  load  must  be  supposed  to  be  furnished 
at  the  end  where  the  current  enters  upon  the  bottom  in  ques- 
tion. In  this  case  deposition  will  first  reduce  the  load  at  the 
end  upon  which  it  enters  and  at  the  same  time  reduce  the  depth 
and  thus  constrict  the  current,  increasing  its  power.  The  latter 
influence  will  determine  a  higher  level  to  which  the  bottom  will 
be  built ;  a  level  at  which  the  power  of  the  water  is  sufficient  to 
carry  the  load  which  before  was  excessive.  Filling  will  then 
advance  forward  over  the  bottom,  the  filled  and  unfilled  portions 
both  being  level,  the  former  growing  while  the  latter  diminishes^ 
and  the  two  being  separated  by  a  slope,  mentioned  below.  It  is 
evident  that  the  depth  at  which  this  slope  begins  is  determined 
jointly  by  the  power  of  the  water,  the  amount  of  the  load,  and 
the  size  of  the  fragments  which  make  up  the  load. 

The  front. — The  shape  of  the  slope  which  intervenes  between 
the  area  which  has  been  filled  and  the  bottom  beyond,  will  be 
determined  by  the  rate  at  which  the  power  of  the  current 
decreases.     If  the  loss  of  power  were  instantaneous,  the  slope 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE      25 

would  be  simply  the  subaqueous  earth  slope.  If  it  be  in  any 
arithmetical  progression,  the  slope  will  be  a  plane  whose  steep- 
ness will  vary  with  the  rate  of  decrease,  the  slope  being  steeper 
when  the  rate  is  higher.  If  the  loss  of  power  be  in  some  other 
manner  than  by  arithmetical  difference,  the  slope  will  show  a 
curve  which  will  be  convex  or  concave  according  as  the  rate  of 
decrease  is  augumented  or  diminished.  In  actual  deposition  by 
a  current  advancing  into  deep  water,  the  decrease  of  power  is  at 
an  increasing  rate,  as  may  be  seen  from  the  following.  If  a  plane 
slope  be  assumed,  so  that  depth  increases  in  arithmetical  ratio, 
then  the  velocity  of  the  current  will  decrease  in  similar  ratio, 
but  transporting  power  varies  as  the  square  of  the  velocity, 
hence  its  rate  of  decrease  is  progressively  augmented.  This  will 
require  convexity  of  slope,  a  feature  generally  observed  at  the 
edge  of  embankments  and  subaqueous  terraces.  The  general 
law  of  equilibrium,  as  given  above  for  an  eroding  current  still 
applies ;  current  power  is  uniform  over  all  parts  of  the  bot- 
tom, if  by  the  term  current  power  is,  understood /<?ze'^;'  with  refer- 
rence  to  load  and  the  current  considered  is  the  resultant  of  all  con- 
flicting currents.  In  this  case,  while  the  current  is  acually  losing 
power,  the  loss  is  balanced  by  the  coincident  loss  of  load,  and 
the  uniformity  of  power  in  comparison  with  load  is  maintained. 
Presence  of  a  bank;  equilibrium  on  a  slope. — The  presence  of  a 
bank  fixes  not  only  a  horizontal  limit  to  the  bottom  in  question, 
but  determines  that  at  this  limit  the  depth  shall  be  zero.  This 
involves  a  slope.  If  equilibrium  is  to  exist  on  this  slope  in  har- 
mony with  the  general  law  stated  above,  the  advantage  in  power 
due  to  shallower  water  on  one  side  must  be  balanced  in  one  of 
four  ways,  (i)  the  equality  of  transporting  power  in  deep  and 
shallow  water  may  be  partially  maintained  by  the  participation 
of  more  water  where  the  depth  is  great  than  where  it  is  small. 
In  the  case  of  undertow  this  has  been  shown  to  be  true;  (2) 
currents  in  both  directions  may  be  stronger,  so  that  the  result- 
ant motion  in  one  direction  may  be  more  in  shallow  water  than 
in  deep  water,  it  may  even  be  zero  or  it  may  be  in  the  opposite 
direction.     The  factors  of  translatory  wave  motion  and  on-shore 


26  A^.  M.  FENNEMAN 

currents  may  occasion  this  condition;  (3)  the  excessive  power 
of  the  water  on  the  shallow  bottom  may  be  employed  in  the 
transporting  of  a  greater  load  or  even  in  erosion.  This  is  quite 
generally  true;  (4)  the  material  may  be  heterogeneous,  the 
larger  stones  coming  to  rest  in  the  shallower  water  because  of 
their  ability  to  withstand  the  greater  agitation  at  a  higher  level. 
Of  all  these  reasons,  it  will  be  seen  that  only  the  first  can  pro- 
vide for  a  permanent  slope  :  the  others  depend  upon  a  continual 
supply  of  fresh  drift. 

Necessity  of  a  continuous  supply  of  load. — Suppose  now  that  a 
short  section  of  coast  line  be  enclosed  between  perfectly  resist- 
ant walls  or  piers  perpendicular  to  the  shore  line,  and  extending 
out  to  deep  water.  The  transportation  of  material  alongshore 
will  thus  be  prevented.  If  the  shore  also  be  supposed  to  be 
perfectly  resistant,  so  that  no  new  drift  can  be  furnished  to  the 
waves,  then  the  profile  of  equilibrium,  toward  which  the  bottom 
will  tend,  is  a  steep  descent  from  the  water  line  to  the  depth  at 
which  undertow  becomes  ineffective,  and  then  a  low  slope  out- 
ward, following  the  base  of  effective  undertow.  This  base  is 
necessarily  on  a  slope  because  of  the  increasing  volume  of  under- 
tow with  distance  from  shore. 

Effect  of  a  supply  of  drift. — If  now,  drift  be  supplied  at  the 
shore  line  at  a  given  rate,  filling  will  occur  at  the  foot  of  the 
steep  descent  leading  down  from  the  water  line,  until  the  bottom 
has  risen  to  a  level  at  which  the  power  of  the  water  is  suflRcient 
to  transport  the  material  at  the  rate  at  which  it  is  furnished,  and 
this  filling  will  advance  off-shore,  ending  in  a  convex  front  as 
shown  above. 

At  the  shoreward  boundary  of  this  filling  area  is  an  angle 
made  by  the  plane  of  deposition,  with  the  steeper  descent  lead- 
ing down  from  the  water's  edge  to  the  line  at  which  deposition 
becomes  possible.  In  an  actual  case,  where  the  material  of  the 
shore  yields  to  erosion,  the  water's  edge  is  carried  landward, 
and  the  first  descent  is  not  only  far  from  vertical,  but  in  weak 
material,  is  very  gentle  ;  probably  always  steeper,  however,  than 
the  slope  made  by  deposition  farther  out.    This  may  be  observed 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE       2/ 

on  almost  any  of  the  coastal  charts  of  the  United  States  Coast 
and  Geodetic  Survey.  The  east  coast  of  Florida  furnishes  typi- 
cal illustrations. 

Normal  profile ;  cutting  coast. — The  normal  profile  then,  of  a 
shore  where  the  resultant  of  transporting  power  is  outward,  is  a 
compound  curve,  which  is  concave  near  the  shore,  passing 
through  a  line  of  little  or  no  curvature,  to  a  convex  front. 
Where  this  front  rests  upon  the  bottom  below  the  reach  of  cur- 
rents, the  descent  merges  into  the  more  level  bottom  by  another 
concave  curve,  due  to  deposition  from  suspension.  If  the  sup- 
ply of  material  from  the  shore  be  cut  off,  the  entire  shelf  will 
be  cut  down  and  its  slope  reduced  and  it  will  necessarily  be 
separated  from  the  shore  by  a  steeper  slope  than  before.  If, 
on  the  other  hand,  the  supply  of  material  be  suddenly  increased, 
a  smaller  shelf  will  grow  from  shore  on  the  surface  of  the  older, 
for  the  reason  that  the  new  load,  being  greater,  is  in  equilibrium 
with  the  currents  at  a  higher  level  than  before.  The  greater  the 
load,  the  nearer  will  the  surface  of  deposition  approach  that  of 
the  water.  On  the  Atlantic  coast  of  the  United  States,  the  depth 
at  which  the  concave  curve  merges  into  the  plane  of  deposition 
varies  from  three  fathoms  near  the  mouths  of  some  rivers,  to  ten 
or  twelve  fathoms  where  the  lead  is  smaller.  On  some  parts  of 
the  Pacific  coast,  where  the  lead  is  small,  the  concave  curve 
descends  to  twenty  or  thirty  fathoms. 

Normal  profile ;  buildmg  coast. — If  the  resultant  of  shore  action, 
be  to  carry  material  landward,  the  general  character  of  the 
resulting  curve  cannot  be  very  different,  since  this  process  also 
produces  steepening  near  shore.  In  general  the  velocity  of 
shoreward  motion  increases  with  nearness  to  land.  If  the 
effectiveness  of  this  motion  increases  with  its  velocity,  there 
is  no  accumulation  until  the  shore  is  reached.  The  shore  is 
then  progressively  steepened  by  accumulation,  until  the  force 
which  acts  shoreward  can  no  longer  carry  material  up  against 
the  growing  component  of  gravity.  This  landward  urging  of 
sediments  is  commonly  thought  to  be  one  of  the  factors  in  the 
production   of  off-shore   barriers.       It   is   plain,   however,   that 


28  iV.il/.  FENNEMAN 

unless  the  power  of  inward  transportation  is  decreased  before 
reaching  the  shore,  no  barrier  can  form.  This  decrease  may, 
at  times  occur,  for  carrying  power  will  depend  not  only  on  the 
velocity,  but  on  the  agitation  of  waves  at  the  bottom.  It  has 
been  seen  that  waves  are  rapidly  reduced  in  size  and  vigor  in 
the  act  of  breaking.  It  is  possible,  therefore,  that  when  the 
slope  is  so  gentle  that  waves  recover  their  form  after  breaking, 
thereby  showing  that  oscillatory  wave  motion  has  been  much 
reduced,  deposition  may  take  place  along  the  line  of  wave  reduc- 
tion, which  is  essentially  the  breaker  line.  With  these  condi- 
tions alone,  however,  the  growth  of  this  feature  would  probably 
be  confined  to  narrow  limits  by  the  undertow.  It  would,  more- 
over, be  a  very  transient  feature,  a  mere  incident  in  the  process 
of  shoreward  transportation.  The  steepening  of  the  shore,  to 
which  this  process  is  incidental,  would  rapidly  remove  the  con- 
ditions of  the  incident. 

Variations  of  the  compound  curve. — The  compound  curve  will 
be  more  marked  in  proportion  as  the  surface  of  deposition  is 
broad  and  its  slope  is  gentle.  Where  it  is  narrow  its  significance 
may  not  appear  from  a  profile  drawn  from  widely  spaced 
soundings.'  If  all  the  waste  from  the  land  be  carried  along- 
shore, the  marginal  terrace  is  of  the  cut  type  purely,  in  which 
the  compound  curve  is  not  noticeable,  the  only  prominent  angle 
being  that  where  the  surface  of  cutting  intersects  the  original 
steeper  bottom. 

Currents  alongshore. —  If  the  effect  of  currents  alongshore 
were  the  same  at  all  distances  from  land,  they  might  be  ignored 
as  a  factor  in  profile  making.  Their  variation  in  strength  at 
different  distances  from  shore  produces  important  results.  It 
has  been  stated  above  that  for  any  o?ie  current  the  power  at 
the  bottom  with  respect  to  the  load  must  remain  constant.  It 
may  also  be  shown  that  of  two  currents^  each  of  which  is  furnished 
with  load  to  its  full  capacity,  the  stronger,  which  may  be  sup- 
posed to  dissipate  gradually,  will  be  in  equilibrium  with  its  load 
at  the  smaller   depth.      Hence   if  transportation  alongshore  be 

*  This  is  illustrated  at  many  places  on  the  Pacific  coast  of  the  United  States. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE      29 

distinctly  greater  in  a  zone  adjacent  to  the  land,  a  smaller 
terrace  will  rest  upon  the  larger.  If  transportation  parallel  to 
the  shore  line  be  distinctly  greater  in  a  zone  off-shore,  and  the 
supply  of  drift  be  at  hand,  a  ridge  will  be  built  along  the  line  of 
this  more  effective  current. 

Barriers. —  It  has  been  shown  above  that  when  the  off-shore 
slope  is  too  low  for  equilibrium,  and  there  are  no  currents 
alongshore,  steepening  is  effected,  in  the  main,  by  accumulation 
at  the  water's  edge,  though  there  may  be  some  small  tendency 
to  accumulation  at  or  just  within  the  breaker  line.  When  cur- 
rents are  flowing,  they  have  a  zone  of  greater  efficiency  along 
this  same  line  or  just  outside.  This  is  because  the  material 
which  they  transport  is  more  agitated  by  wave  action,  and  is  to 
some  extent  lifted  into  the  current.  Excessive  transportation 
along  this  zone  initiates  the  ridge  which  may  continue  to  grow 
until  it  assumes  the  functions  of  the  beach.  It  this  then  called 
a  barrier. 

The  essential  function  of  the  barrier  is  to  steepen  the  bottom 
slope  by  carrying  the  shore  line  farther  out.  If  the  slope  is  not 
abnormally  low,  the  barrier  is  not  needed ;  nor  are  the  con- 
ditions present  which  make  its  formation  possible,  one  of  these 
conditions  being  that  the  agitation  on  the  bottom  at  the  breaker 
line  should  exceed  that  nearer  shore.  It  was  seen  above  that 
this  condition  is  present,  only  on  a  deficient  slope. 

The  slope  may  become  deficient  in  several  ways.  The  cur- 
rents themselves  might  be  the  cause;  or  it  may  result  from  the 
sediments  delivered  by  streams,  as  at  many  places  on  our 
Atlantic  coast ;  or  the  gentle  slope  may  have  belonged  to  the 
original  bottom  over  which  the  waters  rose,  as  seems  to  have 
been  the  case  with  Lake  Michigan  in  its  former  extension  in 
the  vicinity  of  Chicago.  Doubtless  far  the  most  frequent 
occasion  of  deficient  slope  is  the  falling  of  the  water  level  or 
the  rising  of  the  shore.  That  the  immediate  off-shore  slope 
should  in  this  case  be  too  low,  is  the  necessary  consequence  of 
the  concavity  of  the  normal  slope  near  shore.  The  slope  from 
the  Atlantic  shore  line,  where  well  removed  from  rivers,  as  on  the 


30  N.  M.  FENNEMAN 

east  coast  of  Florida,  is  perhaps  ten  fathoms  in  the  first  two 
miles,  but  if  the  sea  level  should  fall  ten  fathoms,  or  the  land 
should  rise  by  that  amount,  the  new  ten-fathom  line  would  lie 
many  miles  off-shore,  and  new  barriers  might  be  expected.  On 
some  of  the  small  lakes  of  Wisconsin,  especially  those  without 
outlet,  as  Silver  Lake  of  the  Oconomowoc  group,  the  falling 
level  has  found  a  deficient  slope  and  barriers  are  constructed. 

The  front  of  the  marginal  shelf. —  If  the  marginal  shelf  be  a 
pure  wave-cut  terrace  with  no  addition  by  deposit,  its  limit  will 
be  marked  by  an  angle  where  the  plane  of  the  shelf  meets  the 
original  bottom.  The  depth  of  the  shelf  at  this  edge  will  con- 
stantly approach  wave-base,  for  it  may  be  safely  assumed  that 
wherever  waves  can  agitate,  there  will  be  sufficient  current  to 
transport.  If  there  are  currents  strong  enough  to  erode  below 
wave-base,  the  shelf  may  be  cut  still  lower.  The  hardness  of 
the  rock  can  make  no  permanent  difference.  This  is  well  illus- 
trated even  in  so  young  and  small  a  body  as  Lake  Mendota 
at  Madison,  Wis.,  where  the  sandstone  shelves  southwest  of 
Governor's  Island  and  Maple  Bluff  are  cut  to  the  same  depth  as 
the  clay  shelves  west  of  Picnic  Point  and  Second  Point. ^ 

If  the  shelf  is  being  broadened  at  the  same  time  by  materials 
carried  across  and  deposited  on  its  front,  there  will  be,  between 
its  upper  surface  and  its  steep  front,  a  curve  convex  to  the  sky 
as  shown  above.  This  steeper  slope  begins,  not  at  the  depth 
where  the  power  of  the  water  ends,  but  at  the  depth  at  which 
the  power  of  the  water  becomes  insuflficient  to  carry  the  entire 
load.  From  this  depth  the  slope  becomes  progressively  steeper 
to  the  depth  at  which  the  movement  of  the  water  is  ineffective. 
Off  the  Atlantic  coast  of  the  United  States,  the  depth  at  which 
the  slope  begins  to  steepen  is  usually  fifty  or  sixty  fathoms,  but 
the  maximum  of  steepness  is  not  attained  until  a  much  greater 
depth  is  reached.  The  depth  familiarly  assigned  to  wave-base 
along  this  coast  is  one  hundred  fathoms,  and  this  figure 
expresses  fairly  well  the  horizon  at  which  the  maximum  steep- 

*  See  hydrographic  map  issued  by  the  Wisconsin  Geological  and  Natural  History 
iSurvey. 


PROFILE  OF  THE  SUBAQUEOUS  SHORE  TERRACE      3 1 

ness  is  reached.  This  would  mean  that  currents  become  unable 
to  carry  the  whole  load  at  fifty  or  sixty  fathoms,  and  at  one 
hundred  fathoms  or  less,  become  unable  to  transport  anything 
except  in  suspension.  If  the  factor  of  transportation  in  suspen- 
sion did  not  enter,  the  front  of  such  a  shelf  should  show  the 
subaqueous  earth-slope. 

It  is  commonly  assumed  as  above,  that  undertow  and  wave 
agitation  lose  their  efficiency  at  the  same  point,  the  limit  of  the 
former  being  determined  by  that  of  the  latter.  Probably  this  is 
very  generally  true  ;  moreover,  since  wave  oscillation  decreases 
with  depth  in  geometrical  ratio  at  a  high  rate,  and  the  decrease 
of  its  agitating  power  is  at  a  rate  measured  by  the  square  of  this 
same  ratio,  it  may  readily  be  seen  that  there  is  a  somewhat  defi- 
nite horizon  below  which  wave  action  is  ineffective.  Such  a 
condition  is  signalized  by  a  somewhat  definite  limit  to  the  sedi- 
mentary shelf. 

Transportation  beyotid  wave-base. — The  undertow  may,  how- 
ever, be  constricted  laterally  and  preserved  from  dissipation,  as 
when  the  water  drifts  into  a  re-entrant  curve  of  the  shore  ;  or 
deep  currents  may  result  from  a  system  of  rebounds.  By  either 
of  these  means  the  power  of  the  lower  water  may  be  increased, 
so  that  at  depths  greater  than  that  of  wave-base  sand  or  even 
gravel  may  be  transported.^  In  such  cases  no  break  in  the  pro- 
file may  be  seen  at  wave-base.  Broad  sheets  or  streaks  of  sand 
may  cover  the  bottom  to  depths  far  beyond  this  line.  Such 
troughs  as  those  of  the  great  lakes,  in  which  all  the  surface 
water  may  be  drifted  simultaneously  in  one  direction,  should 
especially  favor  vertical  circulation  with  vigorous  movements 
below.  Wave-base  of  Lake  Michigan,  where  revealed  by  a 
sharp  angle  at  the  edge  of  a  marginal  terrace,  is  sixty  or  seventy 
feet  below  the  surface ;  yet  around  much  of  its  margin,  a  sand 
covered  or  gravel  covered  bottom,  concave  upward,  extends 
outward  to  several  times  this  depth  with  little  or  no  evidence  of 
change  of  slope  at  wave-base.*     This  is  to  be  expected  from  the 

*  See  H.  C.  Kinahan,  "  The  Beaufort's  Dyke  off  the  coast  of  the  Mull  of  Gal- 
loway," Proceedings  of  the  Royal  Irish  Academy,  Third  Series,  Vol.  VI,  No.  I. 
"  Charts  of  Lake  Michigan,  War  Department. 


32  N.  M.  FENNEMAN 

necessarily  powerful  undertow.  In  Lake  Mendota,  where  wave- 
base  is  not  lower  than  twenty  feet,  sands  and  even  heavy  gravels 
are  irregularly  distributed  over  the  bottom  at  depths  frequently 
approaching  fifty  feet.  Some  lie  at  the  bases  of  steep  slopes 
which  gravity  may  have  helped  them  to  descend,  but  others  are 
far  from  slopes  and  plainly  illustrate  the  erosive  power  of  cur- 
rents resulting  from  a  concentration  of  movement  along  certain 

""^^-  N.  M.  Fenneman. 

The  University  of  Chicago. 


Syracwe,  N.  Y. 
Stockton,  CM. 


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